Vieta Theorem

This exercise includes $13$ intermediate level Vieta Theorem related problems. For example:

(3325) Let $x_1$ and $x_2$ be two real roots of $x^2-x-1=0$. Find the value of $2x_1^5 + 5x_2^3$.

(3327) Let $\alpha$ and $\beta$ be two real roots of $x^4 +k=3x^2$ and also satisfy $\alpha + \beta = 2$. Find the value of $k$.

(3320) Find one real solution $(a, b, c, d)$ to the following system: $$\left\{\begin{array}{rcl}a+b+c+d&=&-2\\ab+ac+ad+bc+bd+cd&=&-3\\abc+abd+acd+bcd&=&4\\abcd&=&3\end{array}\right.$$

Many of these problems come from actual competitions.